Theorem 3netr3d

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 19-Nov-2019)

	Ref	Expression
Hypotheses	3netr3d.1	$$⊢\phi \to A\ne B$$
	3netr3d.2	$$⊢\phi \to A=C$$
	3netr3d.3	$$⊢\phi \to B=D$$
Assertion	3netr3d	$$⊢\phi \to C\ne D$$

Proof

Step	Hyp	Ref	Expression
1		3netr3d.1	$$⊢\phi \to A\ne B$$
2		3netr3d.2	$$⊢\phi \to A=C$$
3		3netr3d.3	$$⊢\phi \to B=D$$
4	1 3	neeqtrd	$$⊢\phi \to A\ne D$$
5	2 4	eqnetrrd	$$⊢\phi \to C\ne D$$